Gregory Chaitin has an article in the March Scientific American
in which he claims that the irreducible complexity of the number
he calls Omega "smashes hopes for a complete, all-encompassing
mathematics in which every true fact is true for a reason."
He also has a popular-level treatment published in book form,
and the book more cautiously notes that his interpretation of
his own work is controversial among philosphers of mathematics.
Any opinions? It doesn't seem surprising to me that there are mathematical
truths that are true, but not "for a reason." E.g., if we take P(n)
to mean that Goldbach's conjecture holds for all numbers up to n,
and Q(n) to mean that the nth digit of Omega is even, then I'd
suspect that for sufficiently large values of n, neither P(n)
not Q(n) can ever be proved or disproved by human efforts, and
neither P(n) nor Q(n) is of any consequence for the rest of
mathematics.